Selectivity in Probabilistic Causality: Drawing Arrows from Inputs to Stochastic Outputs

The problem has applications ranging from modeling pairwise comparisons to reconstructing mental processing architectures to conjoint testing. A necessary and sufficient condition for a given pattern of selective influences is provided by the Joint Distribution Criterion, according to which the problem of "what influences what" is equivalent to that of the existence of a joint distribution for a certain set of random variables. For inputs and outputs with finite sets of values this criterion translates into a test of consistency of a certain system of linear equations and inequalities (Linear Feasibility Test) which can be performed by means of linear programming. The Joint Distribution Criterion also leads to a metatheoretical principle for generating a broad class of necessary conditions (tests) for diagrams of selective influences. Among them is the class of distance-type tests based on the observation that certain functionals on jointly distributed random variables satisfy triangle inequality. A B C The Greek letters in this diagram represent inputs, or external factors, e.g., parameters of stimuli whose values can be chosen at will, or randomly vary but can be observed. The capital Roman letters stand for random outputs characterizing reactions of the system (an observer, a group of observers, a technical device, etc.). The arrows show which factor influences which random output. The factors are treated as deterministic entities: even if α,β,γ,δ in reality vary randomly (e.g., being randomly generated by a computer program, or being concomitant parameters of observations, such as age of respondents), for the purposes of analyzing selective influences the random outputs A, B,C are always viewed as conditioned upon various combinations of specific values of α,β,γ,δ. The first question to ask is: what is the meaning of the above diagram if the random outputs A,B,C in it are not necessarily stochastically independent?